Measurement and simulation strategies for characterizing the response of a given structure to an input, e.g., the electrical behavior of interconnect and packaging structures for electronic circuits, electromagnetic behavior of antenna structures, etc., often utilize a frequency-domain approach. Physically, such an approach is equivalent to applying sinusoidal excitations to the terminals, or ports, of a structure and then measuring the response at the same and/or other ports. The relation between excitations and responses, and how those relations vary with sinusoidal frequency, are used to calculate scattering parameters. For a structure with p ports, measured at nf frequencies (where nf is typically hundreds or thousands), the scattering (or S-) parameter data is a set of nf matrices, where each matrix is a set of p×p complex numbers. The entry in the i-th row and j-th column of the matrix associated with a particular frequency f indicates how a sinusoidal excitation with frequency f applied to port j will affect the response at port i.
Simulators, which are typically used to compute the time evolution of voltages, currents, and/or electromagnetic fields in structures, often require accurate representations of the structure being simulated. For example, circuit simulators typically compute the voltages and/or currents representing the electrical behavior of interconnect and packaging used to couple various circuit blocks. For this reason, almost all commercial circuit simulators have some method for converting models of packaging and interconnect represented using frequency-domain S-parameter data into models that are suitable for time-domain simulation. A wide variety of methods are in common use, with convolution-based approaches being the most established.
The more modern and now preferred strategy for using S-parameter data in time domain circuit simulation is to construct a p-input, p-output system of linear differential equations whose response to sinusoid excitations closely matches the responses represented by the corresponding S-parameter data. Such a system of differential equations, usually referred to as a state-space model, may be easily included in time-domain circuit simulation.
In the case of a structure having a large number of ports, the number of transfer functions (p2, where p denotes the number of ports) is usually much greater than the number of frequencies (nf). This can affect compression, fast residue calculation, fast transfer function calculation during simulation of the structure. Grivet-Talocia describes a singular-value-decomposition-based method of generating a state-space model for structures having a large number of ports. See “A compression strategy for rational macromodeling of large interconnect structures,” S. Grivet-Talocia, S. B. Olivadese, and P. Triverio, Proceedings of the 2011 IEEE 20th Conference on Electrical Performance of Electronic Packaging and Systems (EPEPS), pp. 53-56, 23-26 (October 2011), which is incorporated herein by reference, in its entirety.
The basic idea of this paper is that large port-count S-parameters have substantial redundant information, which the singular value decomposition (SVD) can uncover. A large matrix X is created by stacking the p2 entries of the S-parameter matrix next to each other. Then the singular value decomposition on X is performed, generating matrices U, S, and V. The product of the significant singular values, i.e., entries 1:k of the diagonal of S, with the appropriate columns of U, give the so-called “basis functions” that can represent the important information in the original data. The corresponding columns of V indicate how to combine the basis functions to recreate the original S-parameter data. One problem with Grivet-Talocia's method is that, if the port count p is high, the X matrix is extremely wide, which makes the V matrix extremely tall and significantly slows down the SVD.